The dual of pure non-Abelian lattice gauge theory as a spin foam model

Abstract

We derive an exact duality transformation for pure non-Abelian gauge theory regularized on a lattice. The duality transformation can be applied to gauge theory with an arbitrary compact Lie group G as the gauge group and on Euclidean space-time lattices of dimension d ≥ 2. It maps the partition function as well as the expectation values of generalized non-Abelian Wilson loops (spin networks) to expressions involving only finitedimensional unitary representations, intertwiners and characters of G. In particular, all group integrations are explicitly performed. The transformation maps the strong coupling regime of non-Abelian gauge theory to the weak coupling regime of the dual model. This dual model is a system in statistical mechanics whose configurations are spin foams on the lattice.

Spin Foam Models with Finite Groups

“…Such gauge theories can also be formulated with discrete groups [10,52,90], and for Z 2 , they give the Ising gauge models [10]. A spin foam representation for the Yang-Mills theories with continuous Lie groups has been presented in [91], see also [92,93], and the results can be easily adapted to discrete groups. For the Abelian (discrete and continuous) groups, these are again closely related to the construction of dual models [78] and the high-temperature (or strong coupling) expansion [84].…”

“…Examples for such models are the BarrettCrane model [57] and the EPRL and FK models [58][59][60][61]. These models can also be written in the form of a path integral over connections [91,108,109], but we will not concern ourselves with this here.…”

Generalized lattice gauge theory, spin foams and state sum invariants

“…While this is not a gauge theory anymore it is known to be expressible as a (modified) spin foam model [2]. This dual model was explicitly constructed for hypercubic lattices in [3]. There, it was also shown to be strong-weak dual to the ordinary formulation of LGT.…”

Spin foam model for pure gauge theory coupled to quantum gravity

“…For example, for G = U (1), we have (3.26) where I k denote modified Bessel functions and the irreducible representations are characterized by integers k t ∈ for each triangle t. Similarly for G = SU (2), 27) where ℓ t = 0, are very similar to the four-simplex amplitudes of Figure 1(a), just using the intertwiners attached to the edges incident in v. For more details, see [25,27] where the A…”

“…While the model (3.22) is a hybrid involving a lattice gauge theory together with a spin foam model of gravity, we can make use of the strong-weak duality transformation of lattice gauge theory [25,26,27] in order to obtain a single spin foam model with two types of 'fields'.…”